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Chapter 7, going though after meeting J Howse

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submission_thesis/CH6_Evaluation/copy.tex
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@ -16,14 +16,18 @@ complexity of applying FMEA to a group of components.
These formulae are then used for a hypothetical example, which is analysed by both FMEA and FMMD.
Following on from the formal definitions, `unitary state failure modes' are defined. In short these
ensure that component failure modes are mutually exclusive.
ensure that component failure modes are mutually exclusive. % Using the unitary state failure mode definition
Standard formulae for combinations are then used to develop the concept of
the cardinality constrained power-set. Using this in combination with unitary state failure modes
we can establish an expression for calculated the number of failure scenarios to
check for in double failure analysis.
%
% MOVE TO CH5 FMMD makes the claim that it can perform double simultaneous failure mode analysis without an undue
% MOVE TO CH5 state explosion drawback.
% MOVE TO CH5 To support this, an example of single and double failure analysis is provided, using the four wire Pt100
% MOVE TO CH5 temperature measurement sensor circuit. This example is also used to show how component failure rate statistics can be
% MOVE TO CH5 used with FMMD.
%
This is followed by some critiques i.e. possible areas of difficulty when performing FMMD, and then
a general evaluation. % comparing it with traditional FMEA.
@ -86,7 +90,8 @@ To perform FMEA rigorously
we could stipulate that every failure mode must be checked for effects
against all the components in the system.
%
This would mean we would be looking for all possible side effects that a base component failure could cause.
This would mean we would be %looking
examining for all possible side effects that a base component failure could cause.
%
We could term this `rigorous~FMEA'~(RFMEA).
The number of checks we have to make to achieve this, gives an indication of the complexity of the analysis task.
@ -95,10 +100,11 @@ The number of checks we have to make to achieve this, gives an indication of the
%analyse a single FMEA failure scenario, is given in equation~\ref{eqn:complexity}.
%
%
It is desirable to be able to measure the complexity of an analysis task.
%It is desirable to be able to measure the complexity of an analysis task.
%
Comparison~complexity is a count of
paths between failure modes and components necessary to achieve RFMEA for a given group G. %system or {\fg}.
We define comparison~complexity as the count of
paths between failure modes and components necessary to achieve RFMEA for a given group
of components $G$. %system or {\fg}.
% (except its self of course, that component is already considered to be in a failed state!).
%
@ -121,10 +127,11 @@ $ | G | $. %,
%\paragraph{Defining Components}
$G$ is simply a sub-set of all possible components.
We define the set of all components as $\mathcal{C}$ and can state $G \subset \mathcal{C}$.. Individual components are denoted as $c$
with additional indexing when appropriate.
with additional indexing where appropriate.
\paragraph{Defining a function that returns failure modes given a component.}
The function $fm$ has a component as its domain and the components failure modes, $fms$, as its range. % (see equation~\ref{eqn:fm}).
The function $fm$ has a component as its domain and the components failure modes % , $fms$,
as its range. % (see equation~\ref{eqn:fm}).
Where $\mathcal{F}$ is the set of all failures,
$$ fm: \mathcal{C} \rightarrow \mathcal{F}.$$
We can represent the number of potential failure modes of a component $c$, to be $ | fm(c) | .$
@ -132,12 +139,13 @@ We can represent the number of potential failure modes of a component $c$, to be
\paragraph{Indexing components with the group $G$.}
If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|G|} $ we can express
the number of checks required to rigorously examine every
failure mode against all the other components in the system.
failure mode against all the other components in a system.
Comparison Complexity can be represented by a function $CC$, with its domain as $G$, and
its range as the number of checks---or reasoning stages---to perform to satisfy a rigorous FMEA inspection.
Where $\mathcal{G}$ represents the set of all {\fgs}, and $ \mathbb{Z}^{+} $, $CC$ is defined by,
Where $\mathcal{G}$ represents the set of all {\fgs}%, and $ \mathbb{Z}^{+} $,
$CC$ is defined by,
\begin{equation}
%$$
CC:\mathcal{G} \rightarrow \mathbb{Z}^{+},
@ -146,7 +154,7 @@ Where $\mathcal{G}$ represents the set of all {\fgs}, and $ \mathbb{Z}^{+} $, $C
%
%and, where n is the number of components in the system/{\fg},
and $|fm(c_i)|$ is the number of failure modes
in component ${c_i}$, comparison complexity, $CC$ is given by
in component ${c_i}$, comparison complexity, $CC$ for a group of components $G$, is given by
\begin{equation}
\label{eqn:CC}
@ -158,7 +166,6 @@ in component ${c_i}$, comparison complexity, $CC$ is given by
This can be simplified if we can determine the total number of failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$);
equation~\ref{eqn:CC} becomes
%$$
\begin{equation}
\label{eqn:rd2}
@ -171,7 +178,7 @@ We define the set of all {\fgs} as $\mathcal{FG}$.
Using $FG$ to represent individual {\fgs} we %can therefore
state $$ \forall FG \in \mathcal{FG} | FG \subset \mathcal{G} .$$
FMMD analysis creates a hierarchy $H$ of {\fgs} where $H \subset \mathcal{FG}$.
FMMD analysis creates a hierarchy $\hh$ of {\fgs} where $\hh \subset \mathcal{FG}$.
%
We can define individual {\fgs} using $FG^{\alpha}_{i}$ with an index, $i$ for identification and a superscript for the $\alpha$~level (see section~\ref{sec:alpha}).
%---
@ -192,26 +199,32 @@ i.e. at the zeroth level of an FMMD hierarchy where $\alpha=0$, would have the s
An FMMD Hierarchy will have reducing numbers of {\fgs} as we progress up the hierarchy.
In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
all {\fgs} on each level.
We can define an FMMD hierarchy as a set of {\fgs}, $H$.
We define a helper function $g$ with a domain of the level $i$ in an FMMD hierarchy $H$, and a co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level),
defined by,
We can define an FMMD hierarchy as a set of {\fgs}, $\hh$.
We define a helper function $g$ with a domain of the level $Level$ in an FMMD hierarchy $\hh$, and a
co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level),
that returns
the sum of all complexity comparison
applied to {\fgs} at a particular hierarchy level in \hh,
\begin{equation}
%$$
g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\xi} \in H) .
%$$
g: \hh \times \mathbb{N} \rightarrow \mathbb{N} .
\end{equation}
IN ENGLISH: A helper function $g$ that returns all {\fgs} at a particular hierarchy level in a particular FMMD hierarchy.
%$$
%g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\xi} \in H) .
%$$
Where $L$ represents the number of levels in the FMMD hierarchy,
$|g(\xi)|$ represents the number of {\fgs} on the level
and $H$ represents an FMMD hierarchy,
we overload the comparison complexity thus:
%IN ENGLISH: A helper function $g$
%
Where $L$ represents the number of levels in the FMMD hierarchy {\hh} and
$g(\hh,\xi)$ represents the comparison complexity of {\fgs} on the level $\xi$;
%and $\hh$ represents an FMMD hierarchy,
we overload the comparison complexity function $CC$, to obtain the comparison complexity of an entire hierarchy thus:
%$$
\begin{equation}
\label{eqn:gf}
CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|g(H,\xi)|} CC({\FG}_{j}^{\xi}).
%% CC(\hh) = \sum_{\xi=0}^{L} \sum_{j=1}^{|g(\hh,\xi)|} CC({\FG}_{j}^{\xi}).
CC(\hh) = \sum_{\xi=0}^{L} g(\hh,\xi).
%$$
\end{equation}
@ -225,7 +238,7 @@ $$CC(invamp) = 2 \times 1 + 4 \times 1 = 6 $$
To analyse the inverting amplifier with FMMD we required 10 reasoning stages.
Using RFMEA we obtain $ 2 \times (3-1) + 2 \times (3-1) + 4 \times (3-1)$ = 16.
\paragraph{Complexity Comparison for an 81 component system.}
\paragraph{Complexity Comparison for an hypothetical 81 component system.}
%Even considering a $example$
A system, $example$, with just 81 components (with these components
having 3 failure modes each) we would have an $CC$ of
@ -236,7 +249,7 @@ Ensuring all component failure modes are checked against all other components in
-- applying FMEA rigorously -- could be termed
Rigorous FMEA (RFMEA).
The computational order for RFMEA would be polynomial ($O(N^2.K)$) (where $K$ is the variable number of failure modes).
%
This order may be acceptable in a computational environment: However, the choosing of {\fgs} and the analysis
process are by-hand/human activities. It can be seen that it is practically impossible to achieve
RFMEA for anything but trivial systems.
@ -262,7 +275,7 @@ rigorous checking feasible.
\centering
\includegraphics[width=400pt]{./CH6_Evaluation/components_81_euler.png}
% components_81_euler.png: 3056x2532 pixel, 72dpi, 107.81x89.32 cm, bb=0 0 3056 2532
\caption{FMMD Hierarchy with number of components in each $FG$ fixed to three ($|FG|=3$)}
\caption{Euler diagram of a hypothetical FMMD Hierarchy with 81 base components with the number of components in each $FG$ fixed to three ($|FG|=3$)}
\label{fig:three_tree}
\end{figure}
@ -310,7 +323,7 @@ three failure modes.
Thus the number of checks to make in the top level is $3^0\times3\times2\times3 = 18$.
%
On the level below that, we have three {\fgs} each with
an identical number of checks, $3^1 \times 3 \times 2 \times 3 = 56$.%{\fg}
an identical number of checks, $3^1 \times 3 \times 2 \times 3 = 56$. %{\fg}
%
On the level below that we have nine {\fgs}, $3^2 \times 3\times2\times3=168$.
Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA {\em{within the}}
@ -635,10 +648,10 @@ Thus if the failure modes of a component $F$ are unitary~state, we can say $F \
\section{Component failure modes: Unitary State example}
An example of a component with an obvious set of ``unitary~state'' failure modes is the electrical resistor.
Electrical resistors can fail by going OPEN or SHORTED.
For a given resistor R we can apply the
%
We use the EN298~\cite{en298}[Ann.A] failure mode definition for resistors: OPEN or SHORTED.
%
For a given resistor R we could apply the
function $fm$ to find its set of failure modes thus $ fm(R) = \{R_{SHORTED}, R_{OPEN}\} $.
A resistor cannot fail with the conditions open and short active at the same time,
that would be physically impossible! The conditions
@ -648,24 +661,25 @@ Because of this, the failure mode set $F=fm(R)$ is `unitary~state'.
%
%Thus because both fault modes cannot be active at the same time, the intersection of $ R_{SHORTED} $ and $ R_{OPEN} $ cannot exist.
%
The intersection of these is therefore the empty set, $ R_{SHORTED} \cap R_{OPEN} = \emptyset $,
The intersection of these failure modes is therefore the empty set, $ R_{SHORTED} \cap R_{OPEN} = \emptyset $,
therefore
$ fm(R) \in \mathcal{U} $.
$ fm(R) \in \mathcal{U} $. These concepts are expanded in section~\ref{sec:usprob}.
We can make this a general case by taking a set $F$ (with $f_1, f_2 \in F$) representing a collection
of component failure modes.
%
We can define a Boolean function {\ensuremath{\mathcal{ACTIVE}}} that returns
whether a fault mode is active (true) or dormant (false).
%
We can say that if any pair of fault modes is active at the same time, then the failure mode set is not
unitary state:
we state this formally
we state this formally;
\begin{equation}
\exists f_1,f_2 \in F \dot ( f_1 \neq f_2 \wedge \mathcal{ACTIVE}({f_1}) \wedge \mathcal{ACTIVE}({f_2}) ) \implies F \not\in \mathcal{U}
\exists f_1,f_2 \in F \dot ( f_1 \neq f_2 \wedge \mathcal{ACTIVE}({f_1}) \wedge \mathcal{ACTIVE}({f_2}) ) \implies F \not\in \mathcal{U} .
\end{equation}
@ -687,7 +701,10 @@ we have banned larger combinations as well.
All components must have unitary state failure modes to be used with the FMMD methodology and
for base~components this is usually the case. Most simple components fail in one
clearly defined way and generally stay in that state.
clearly defined way and generally stay in that state.
Traditional FMEA has problems dealing with non unitary state failure modes.
This is mainly because combinations of failure modes could cause
effects very difficult to predict (as they are in effect new failure modes of the component).
However, where a complex component is used, for instance a microcontroller
with several modules that could all fail simultaneously, a process
@ -707,7 +724,7 @@ is then applied to it.}.
\paragraph{Reason for Constraint.} Were this constraint not to be applied
\paragraph{Reason for FMMD unitary failure mode constraint.} Were this constraint not to be applied
each component would not contribute $N$ failure modes, % to consider
but potentially
$2^N$.
@ -813,7 +830,7 @@ calculation (in equation \ref {eqn:ccps}) would give the correct number of test
Because sets of failure modes in FMMD analysis are constrained to be unitary state,
the actual number of test cases to check will usually
be less than this.
This is because combinations of faults within a components failure mode set
This is because certain combinations of faults within a components failure mode set
are impossible under the conditions of unitary state failure mode.
To modify equation \ref{eqn:ccps} for unitary state conditions, we must subtract the number of component `internal combinations'
for each component in the functional group under analysis.
@ -922,7 +939,8 @@ associated with the test cases, complete coverage would be verified.
We use the Pt100 example in~\ref{sec:Pt100} which performs double failure mode FMMD analysis.
It is important to check that we have covered all possible double fault combinations.
We can use the equation \ref{eqn:correctedccps2}
We can use the equation \ref{eqn:correctedccps2} to determine the number of failure scenarios, or checks,
we should have made for complete failure coverage.
\ifthenelse {\boolean{paper}}
{
from the definitions paper
@ -941,7 +959,7 @@ reproduced below to verify this.
}
\begin{equation}
|{\mathcal{P}_{cc}SU}| = {\sum^{k}_{1..cc} \frac{|{SU}|!}{k!(|{SU}| - k)!}}
- {{\sum^{j}_{j \in J} \frac{|FM({C_j})|!}{2!(|FM({C_j})| - 2)!}} }
- {{\sum_{j \in J} \frac{|FM({C_j})|!}{2!(|FM({C_j})| - 2)!}} }
\label{eqn:correctedccps2}
\end{equation}
@ -965,7 +983,7 @@ Populating this equation with $|SU| = 6$ and $|FM(C_j)|$ = 2.
\begin{equation}
|{\mathcal{P}_{2}SU}| = {\sum^{k}_{1..2} \frac{6!}{k!(6 - k)!}}
- {{\sum^{j}_{1..3} \frac{2!}{p!(2 - p)!}} }
- {{\sum_{1..3} \frac{2!}{2!(2 - 2)!}} }
%\label{eqn:correctedccps2}
\end{equation}
@ -1036,6 +1054,7 @@ in the Pt100 circuit.
\pagebreak[1]
\section{Component Failure Modes and Statistical Sample Space}
\label{sec:usprob}
%\paragraph{NOT WRITTEN YET PLEASE IGNORE}
A sample space is defined as the set of all possible outcomes.
For a component in FMMD analysis, this set of all possible outcomes is its normal (or `correct')

3
submission_thesis/style.tex
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@ -5,6 +5,8 @@
\DeclareMathSymbol{\Q}{\mathbin}{AMSb}{"51}
\DeclareMathSymbol{\I}{\mathbin}{AMSb}{"49}
\DeclareMathSymbol{\C}{\mathbin}{AMSb}{"43}
%\DeclareMathSymbol{\hh}{\mathbin}{AMSb}{"48}
\newcommand{\ft}{\ensuremath{4\!\!\rightarrow\!\!20mA} }
\usepackage{graphicx}
@ -34,6 +36,7 @@
\newcommand{\sd}{\ensuremath{\Sigma \Delta ADC}}
%\newcommand{\sd}{\ensuremath{Sigma\;Delta\;ADC}}
\newcommand{\derivec}{{D}}
\newcommand{\hh}{\ensuremath{{\stackrel{o}{H}}}}
\newcommand{\abslev}{\ensuremath{\alpha}}
\newcommand{\oc}{\ensuremath{^{o}{C}}}
\newcommand{\adctw}{{${\mathcal{ADC}}_{12}$}}